   Chapter 9.7, Problem 13E ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042

#### Solutions

Chapter
Section ### Mathematical Applications for the ...

11th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781305108042
Textbook Problem

# Find the derivatives of the functions in Problems 1-32. Simplify and express the answer using positive exponents only. y = 5 3 x 3 ( 4 x 5 − 5 ) 3

To determine

To calculate: The simplified form of the derivative of y=53x3(4x55)3.

Explanation

Given Information:

The function is y=53x3(4x55)3.

Formula used:

According to the power rule, if f(x)=xn, then,

f(x)=nxn1

According to the property of differentiation, if a function is of the form, g(x)=cf(x), then,

g(x)=cf(x)

According to the property of differentiation, if a function is of the form f(x)=u(x)+v(x), then,

f(x)=u(x)+v(x)

According to the product rule, if f(x)=u(x)v(x), then

f(x)=u(x)v(x)+v(x)u(x)

The derivative of a constant value, k is

ddx(k)=0

According to the property of differentiation, if a function is of the form y=un, where u=g(x),

dydx=nun1dudx

Calculation:

Consider the provided function,

y=53x3(4x55)3

Consider (4x55) to be u,

y=53x3u3

Differentiate both sides with respect to x,

y=ddx(53x3u3)=53ddx(x3u3)

Simplify by the use of the product rule,

y=53((ddx(x3))u3+(ddx(u3))x3)

Simplify by the use of the power rule,

y=53((3x31)u3+(3u31dudx)x3)=53((3x2)u3+(3u2dudx)x3

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